An Asset Pricing Model with Biases in Beliefs
42 Pages Posted: 14 Mar 2006
Date Written: March 2006
Abstract
The Capital Asset Pricing Model (CAPM) is based on the Expected Utility (EU) theory that assumes linearity in probabilities and models the assumption of risk-aversion via a concave utility function. Studies in psychology and economics, however, show that individuals systematically violate the assumption of linearity in probabilities and this affects their risk attitudes. I derive a measure of risk-premium that arises due to non-linearity in probabilities, captured by a non-linear probability weighting function, and arrive at a measure of conditional risk-aversion. I show that properties of the weighting function implies that the measure of conditional risk-aversion is decreasing in the outcome, in the case of a gamble with a uniform distribution. For the case of a normal distribution, I describe sufficient conditions on the distribution under which the measure of conditional risk-aversion is decreasing in the outcome. Further, I make a set of plausible assumptions on the weighting function and the measure of conditional risk-aversion, to reflect an individual's probability beliefs about a gamble; and characterize the shape of implied marginal utility function in the EU framework. In an asset-pricing model, such probability beliefs implies that the representative agent behaves as if risk-seeking in some regions of outcome, in particular, the small decumulative probability region of positive excess market returns (corresponding to high market gains). Moreover, the stochastic discount factor (SDF) is increasing in, among possibly other regions, the small decumulative probability region of positive excess market returns. This is in contrast to a decreasing SDF implied by the CAPM and its extensions. The model's prediction for the SDF, and consequently, its implications for the cross-section of expected returns, provides a plausible explanation for the weak relationship between a stock's average return and its CAPM beta.
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